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PATTERNS AND MODELS

James Mariner

OBJECTIVES

  1. To recognize patterns in objects and events
  2. To construct models about objects and events that explain observations
  3. To test and accept, modify, or reject models to improve their representation of reality

MATERIALS

  • OB-sertainer KitĘ (available from Sargent -Welch/VWR Scientific)
  • Black Box Kit ("sealed black box" w/ 2-3 skewers in each direction on which 3-6 small washers have been strung; available from Hubbard)

BACKGROUND

We are surrounded by patterns. The day has a pattern of light and dark. Our activities even exhibit patterns that relate to the daily pattern. Certainly, we do different things on different days, but the general patterns are often quite similar -- mealtimes, sleep times, class times, etc. The year itself has a pattern of spring, summer, fall, and winter. Patterns exist in our language -- words and letters that always go together, and others that never do, for example.

Our numbering system has a pattern that repeats itself every ten counting units. This is why we say that it is in the base 10. Another counting system, used in computers, has only two digits. We say that it is a binary system because it uses the base 2:

    Can you finish counting to 10 in the binary system?

    Some patterns are easy to see and describe; others can be quite obscure (cryptic). With some investigations, patterns that at first might seem to be rather obscure can be discovered. For example, when you came to class today, you became aware that you were seated in the room by some specific pattern that the teacher had in mind. Can you discover the pattern involved? Describe in detail the manner in which you tested your ideas.




    The first hypotheses that you developed to explain the seating arrangement might be called models. They were ideas that you had suggested to explain your discomfort with the new seating arrangement. Models are not necessarily right, but they do explain observations if they are good models. Some models may be good for a while, but more information may cause them to be modified or even discarded entirely in favor of new ones. In this series of exercises, we will be using models to explain and predict.

Exercises
  1. Look at the following sequence of numbers:

    13, 14, 16, 19

    1. What would you expect the next number to be?

    2. Explain your answer in (a).





  2. Look at the following number series:

    13 + 14 + 16 + 19 + 13 + 14 + 16 + 19 + 13 + 14 + 16 + 19

    1. What is the sum of the numbers? [Hint: look for a pattern that will give you a short-cut to the solution.]

    2. Describe how you arrived at your answer:

  3. Find a quick and easy method for finding the sum of the first 100 counting numbers (w/o calculators):

    1 + 2 + 3 + 4 + 5 + ... + 95 + 96 + 97 + 98 + 99 + 100

    1. What is your answer?

    2. Describe the pattern that provides your short-cut:

  4. How about the sum of all of the EVEN numbers between 0 and 100 (including 100)?

    1. Your answer:

    2. Describe your solution:

  5. Look at the following pattern of dots:
    1. How many dots are in the 5th term?
      Draw the pattern in the space provided above.

    2. What is the value of the 10th term?
    3. Describe the pattern you have discovered to predict the value of any given (nth) term:






    4. Express this pattern as a formula using T as the total number of dots in a given term and n as the term.

      • T =

  6. Look at the following sequence of crosses and complete the table:
    1. What is the value of the 5th term?
      Draw the pattern in the space provided above.

    2. What is the value of the 10th term?
    3. Describe the pattern you have discovered to predict the value of the nth term:


    4. Try to express this pattern as a formula using T as the total number of dots in a given term and n as the term.

      T =

    The Fibonacci Series is a well-known sequence of numbers: . . . . n:
    Sequence: 1, 1, 2, 3, 5, 8, 13.....
    1 2 3 4 5 6 7

    1. What should the 8th number in the sequence be?
    2. Describe any patterns you observe:


    3. Compare the sum of the first 3 terms with the 5th term. Does this pattern hold for the sum of the first 4 terms? the first 6 terms?


      Support your answer by showing two other specific examples.




  7. Obtain one of the blue "Obsertainers" from your teacher. Within it is a small ball bearing that is free to roll except for an inner molded pattern. Without opening the Obsertainer, see if you can determine the shape and position of the inner pattern. Draw it in the circle (a), below. Then try a few more.

  8. In the same manner and with a partner, try your hand at a "Black Box". First, make a drawing for the arrangement of washers on the skewers that you think is reasonable. Complete the hypothesis below to guide you in the testing of your idea. Then, carefully remove the skewer that will test your hypothesis. Does it confirm or deny your hypothesis? Continue your testing for the arrangement of washers within the box, revising your hypothesis as necessary.

    Hypothesis #1: If the washers are arranged as shown in the diagram at the right, and I remove skewer _____, then...

    Result:

    Hypothesis #2:


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